Lessons from Nanoscience: A Lecture Note Series · I: The New Perspective 15 2. Why Electrons Flow...

Lessons from Nanoscience: A Lecture Note Series

Volume 1: Lessons from Nanoelectronics:

A New Perspective on Transport

Supriyo Datta Purdue University [email protected]

https://nanohub.org/groups/lnebook

World Scientific Publishing Company First Edition 2012, Second Edition 2015 (to appear)

Copyright Supriyo Datta

Contents

Revised Manuscript for II Edition

1. Overview Video Lecture 04

Part A: Basic Concepts I: The New Perspective 15

2. Why Electrons Flow 17 3. The Elastic Resistor 31 4. Ballistic and Diffusive Transport 41 5. Kubo Formula* 49 Appendix A. Fermi and Bose Function Derivatives 57 Appendix B. Angular Averaging 59 Further Reading 61

II: Energy Band Model 63

6. Conductivity 65 7. The Nanotransistor 83

MATLAB Codes* 100 Further Reading 104

III: What and Where is the Voltage? 105

8. Diffusion Equation for Ballistic Transport 107 9. Electrochemical Potentials and Quasi-Fermi levels 121

10. Boltzmann Equation 141 11. Hall Effect* 155 12. Spin Voltages 167

Appendix C. Hamiltonian with E- and B-Fields* 189 Appendix D. Transmission Line Parameters from BTE* 191

Further Reading 194 * indicates chapters not discussed in this course

https://www.youtube.com/watch?v=zaTlwqW0QDA

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IV: Electricity & Heat 197

13. Thermoelectricity 199 14..Phonon Transport* 219 15. Second Law 229 16. Fuel Value of Information 251 Further Reading 264

17. Summary and Looking Forward : 267

From Semiclassical to Quantum

Part B: Quantum Transport* (Covered in the next course, not in this one)

V: Schrodinger Equation

18. Hamiltonian matrix

VI: Contact-ing Schrodinger 19. Non-Equilibrium Green’s Functions (NEGF) 20. Can Two Offer Less Resistance than One? 21. Quantum of Conductance 22. Inelastic scattering

23. Does NEGF Include “Everything”?

VII: Spin Transport 24. Rotating an Electron 25. Spin Circuits

Epilogue: Spin Transistor and Beyond

References / Further Reading

Appendices

E. NEGF Equations F. NEGF from Second Quantized Formalism G. MATLAB Codes for Text Figures

* indicates chapters not discussed in this course


Some Symbols Used

Constants

Electronic charge - q - 1.6e-19 coul. Unit of Energy 1 eV + 1.6e-19 Joules

Planck's constant h 6.626e -34 Joule-sec 1.055e -34 Joule-sec

Boltzmann constant k 1.38e-23 Joule / K ~ 25 meV / 300K

Free electron mass m0 9.11e-31 Kg Effective mass m

Other Symbols I Electron Current amperes (A)

(See Fig.3.2) V Electron Voltage volts (V) U Electrostatic Potential eV µ Electrochemical Potential eV (also called Fermi level or quasi-Fermi level) µ0 Equilibrium eV

Electrochemical Potential

R Resistance Ohms (V/A) G Conductance Siemens (A/V) G(E) Conductance at 0K with µ0=E Siemens (A/V)

Diffusivity m2 /sec Mobility m2 /V-sec Resistivity Ohm-m (3D), Ohm (2D) Conductivity S/m (3D), S (2D)

= h / 2π

D

µρ

σ

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A Area m2

W Width m

L Length m

E Energy eV f (E) Fermi Function Dimensionless

−∂ f∂E

⎛⎝⎜

⎞⎠⎟

Thermal Broadening Function (TBF) / eV

kT −∂ f∂E

⎛⎝⎜

⎞⎠⎟

Normalized TBF Dimensionless

D(E) Density of States /eV N(E) Number of States with Energy < E Dimensionless

Equals Number of Electrons at 0K with µ0=E n Electron Density (3D or 2D or 1D) /m3 or /m2 or /m M(E) Number of Channels Dimensionless

(also called transverse modes) T Temperature degrees Kelvin (K) t Transfer Time seconds

PART B: Quantum Transport ν Transfer Rate /second γ ≡ ν Energy Broadening eV

[X]+ Complex conjugate of transpose of matrix [X] H (Matrix) Hamiltonian eV GR(E) (Matrix) Retarded Green’s function /eV GR(E) = [GA(E)]+ (Matrix) Advanced Green’s function /eV Gn(E) / 2π (Matrix) Electron Density /eV, per gridpoint

A(E) / 2π (Matrix) Density of States /eV, per gridpoint Γ(E) (Matrix) Energy Broadening e

3

Chapter One

Overview

Overview Lecture Video “Everyone” has a computer these days, and each computer has more than a billion transistors, making transistors more numerous than anything else we could think of. Even the proverbial ants, I am told, have been vastly outnumbered. There are many types of transistors, but the most common one in use today is the Field Effect Transistor (FET), which is essentially a resistor consisting of a “channel” with two large contacts called the “source” and the “drain” (Fig. 1.1a). Fig.1.1a. The Field Effect Transistor (FET) is essentially a resistor consisting of a “channel” with two large contacts called the “source” and the “drain”, across which we attach the two terminals of a battery.

The resistance R = Voltage (V) / Current (I) can be switched by several orders of magnitude through the voltage VG applied to a third terminal called the “gate” (Fig.1.1b) typically from an “OFF” state of ~100 megohms to an “ON” state of ~10 kilohms.

4 Lessons from Nanoelectronics


Fig.1.1b. The resistance R = V/I can be changed by several orders of magnitude through the gate voltage VG.

Actually, the microelectronics industry uses a complementary pair of transistors such that when one changes from 100M to 10K, the other changes from 10K to 100M. Together they form an inverter whose output is the "inverse" of the input: A low input voltage creates a high output voltage while a high input voltage creates a low output voltage as shown in Fig.1.2. A billion such switches switching at GHz speeds (that is, once every nanosecond) enable a computer to perform all the amazing feats that we have come to take for granted. Twenty years ago computers were far less powerful, because there were “only” a million of them, switching at a slower rate as well. Fig.1.2. A complementary pair of FET’s form an inverter switch.

Overview 5

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Both the increasing number and the speed of transistors are consequences of their ever-shrinking size and it is this continuing miniaturization that has driven the industry from the first four-function calculators of the 1970’s to the modern laptops. For example, if each transistor takes up a space of say 10 µm x 10 µm, then we could fit 3000 x 3000 = 9 million of them into a chip of size 3cm x 3cm, since That is where things stood back in the ancient 1990’s. But now that a transistor takes up an area of ~ 1 µm x 1 µm, we can fit 900 million (nearly a billion) of them into the same 3cm x 3cm chip. Where things will go from here remains unclear, since there are major roadblocks to continued miniaturization, the most obvious of which is the difficulty of dissipating the heat that is generated. Any laptop user knows how hot it gets when it is working hard, and it seems difficult to increase the number of switches and/or their speed too much further. These Lectures, however, are not about the amazing feats of microelectronics or where the field might be headed. They are about a less-appreciated by-product of the microelectronics revolution, namely the deeper understanding of current flow, energy exchange and device operation that it has enabled, based on which we have proposed what we call the bottom-up approach. Let me explain what we mean. According to Ohm's law, the resistance R is related to the cross-sectional area A and the length L by the relation

(1.1a)

ρ being a geometry-independent property of the material that the channel is made of.

�

3 cm /10 µm = 3000

R ≡V

I=

ρL

A



The reciprocal of the resistance is the conductance

I

V=

σ A

L (1.1b)

which is written in terms of the reciprocal of the resistivity called the conductivity. Our conventional view of electronic motion through a solid is that it is "diffusive," which means that the electron takes a random walk from the source to the drain, traveling in one direction for some length of time before getting scattered into some random direction as sketched in Fig.1.3. The mean free path, λ that an electron travels before getting scattered is typically less than a micrometer (also called a micron = 10-3 mm, denoted µm) in common semiconductors, but it varies widely with temperature and from one material to another. Fig.1.3. The length of the channel of an FET has progressively shrunk with every new generation of devices (“Moore’s Law”) and stands today (2010) at ~ 50 nm, which amounts to a few hundred atoms!

Overview 7

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It seems reasonable to ask what would happen if a resistor is shorter than a mean free path so that an electron travels ballistically ("like a bullet") through the channel. Would the resistance still obey Ohm's law? Would it still make sense to talk about its resistance? These questions have intrigued scientists for a long time, but even twenty five years ago one could only speculate about the answers. Today the answers are quite clear and experimentally well established. Even the transistors in commercial laptops now have channel lengths L ~ 50 nm, corresponding to a few hundred atoms in length! And in research laboratories people have even measured the resistance of a hydrogen molecule.

It is now clearly established that the resistance of a ballistic conductor can be written in the form

RB =h

q2

~ 25 KΩ

1

M (1.2)

where h/q2 is a fundamental constant and M represents the number of effective channels available for conduction. Note that here we are using the word “channel” not to denote the physical channel in Fig.1.3, but in the sense of parallel paths whose meaning will be clarified in the next few lectures. In future we will refer to M as the number of “modes”. This result is now fairly well-known, but the common belief is that it applies only to short conductors and belongs in a course on special topics like mesoscopic physics or nanoelectronics. What is not as well-known is that the resistance for both long and short conductors can be written in the form (λ : mean free path)

R =h

q2M

RB

1+L

λ⎛⎝⎜

⎞⎠⎟

(1.3)



Ballistic and diffusive conductors are not two different worlds, but rather a continuum as the length L is increased. For L << λ , Eq.(1.3) reduces to the ballistic result in Eq.(1.2), while for L >> λ , it morphs into Ohm’s law in Eq.(1.1). Indeed we could rewrite Eq.(1.3) in the form

R =ρ

AL + λ( ) (1.4)

with the resistivity given by

ρ = hq2

AM

1λ

(1.5a)

This expression for resistivity provides a different perspective compared to the standard picture conveyed by the Drude formula taught in every freshman physics text:

ρ = mq2 n

1τ

(Drude formula) (1.5b)

where the resistivity involves the effective mass (m), the number of free electrons (n) and the mean free time (τ ). In our new perspective (Eq.(1.5a)) the resistivity involves the number of modes per unit area (M/A) and the mean free path (λ ).

This new perspective illustrates the essence of our bottom-up approach, viewing short conductors not as an aberration but as the starting point to understanding long conductors. For historical reasons, the subject of conduction is always approached top-down, from large complicated conductors down to hydrogen molecules. As long as there was no experimental evidence for what the resistance of a small conductor might be, it made good sense to start from large conductors where the answers were clear. But now that the answers are clear at both ends, a bottom-up view seems called for, at least to complement the top-down view. After all that is how we learn most things, from the simple to the complex:

Overview 9

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quantum mechanics, for example, starts with the hydrogen atom, not with bulk solids. But there is a deeper reason why the bottom-up approach can be particularly useful in transport theory and this is the “new perspective” we are seeking to convey in these lectures. One of the major conceptual issues posed by the ballistic resistance RB in Eq.(1.2), is the question of “where is the heat”. Current flow through any resistance R leads to the generation of an amount of heat VI = I2R, commonly known as Joule heating. A ballistic resistance RB too must generate a heat of I2RB. But how can a ballistic resistor generate heat? Heat generation requires interactions whereby energetic electrons give up their excess energy to the surrounding atoms. A conductor through which electrons zip through without exchanging energy cannot possibly be generating any heat. It is now generally accepted that in such a resistor, all the Joule heat would be dissipated in the contacts as sketched in Fig.1.4. There is experimental evidence that real nanoscale conductors do approach this ideal and a significant fraction of the Joule heat is generated in the contacts. Fig.1.4. The ideal elastic resistor with the Joule heat VI = I2R generated entirely in the contacts as sketched. Many nanoscale conductors are believed to be close to this ideal. In a sense this seems obvious as my colleague Ashraf often points out. After all a bullet dissipates most of its energy to the object it hits, rather than to the medium it flies through. And yet in the present context, this does seem like a somewhat counter-intuitive result. Clearly the flow of electrons and hence the resistance is determined by the area of the narrow channel that electrons have to squeeze through and not by the



large area contacts. But the associated Joule heat occurs in the contacts. And this would be true even if the channel were full of “potholes” that scattered the electrons, as long as the interaction with the electrons is purely elastic, that is does not involve any transfer of energy, The point we wish to make is that flow or transport always involves two fundamentally different types of processes, namely elastic transfer and heat generation, belonging to two distinct branches of physics. The first involves frictionless mechanics of the type described by Newton's laws or the Schrödinger equation. The second involves the generation of heat described by the laws of thermodynamics. The first is driven by forces or potentials and is reversible. The second is driven by entropy and is irreversible. Viewed in reverse, such processes look absurd, like heat flowing spontaneously from a cold to a hot surface or an electron accelerating spontaneously by absorbing heat from its surroundings. Normally the two processes are intertwined and a proper description of current flow in electronic devices requires the advanced methods of non-equilibrium statistical mechanics that integrate mechanics with thermodynamics. Over a century ago Boltzmann taught us how to combine Newtonian mechanics with heat generating or entropy-driven processes and the resulting Boltzmann transport equation (BTE) is widely accepted as the cornerstone of semiclassical transport theory. The word semiclassical is used because some quantum effects have also been incorporated approximately into the same framework.

Overview 11

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A full treatment of quantum transport requires a formal integration of quantum dynamics described by the Schrodinger equation with heat generating processes. This is exactly what is achieved in the non-equilibrium Green function (NEGF) method originating in the 1960’s from the seminal works of Martin and Schwinger (1959), Kadanoff and Baym (1962), Keldysh (1965) and others (see Chapter 19). The BTE takes many semesters to master and the full NEGF formalism, even longer. Much of this complexity, however, comes from the difficulty of combining mechanics with distributed heat-generating processes. The operation of the elastic resistor can be understood in far more elementary terms because of the clean spatial separation between the force-driven and the entropy-driven processes. The former is confined to the channel and the latter to the contacts. As we will see in the next few lectures, the latter is easily taken care of, indeed so easily that it is easy to miss the profound nature of what is being accomplished. Even quantum transport can be discussed in relatively elementary terms using this viewpoint. My own research has largely been focused in this area developing the NEGF method, but we will get to it only in Part B after we have “set the stage” in Part A using a semiclassical picture. But does this viewpoint help us understand long conductors? Short conductors may be quasielastic and conceptually simple, but don’t we finally have to deal with distributed heat generation if we want to understand long conductors?



We would like to argue that we can use the elastic resistor in Fig.1.4 to understand long conductors from a perspective that is very different from the standard one. The key insight is that what gives rise to a resistance R and where the corresponding heat I2R is dissipated are separate things. For example, if we make a hole in the middle of the channel, the resistance R will undoubtedly go up but the heat I2R clearly has to be dissipated somewhere else, since that requires energy to be transferred from the electrons to the atoms to set them jiggling. A hole does not have the degrees of freedom needed to absorb energy. Fig.1.5. A hole in the middle of the channel gives rise to a large resistance R, but does not have the degrees of freedom needed.to dissipate the corresponding Jould heat I2R. The resistance R and the resulting Joule heat I2R involve different physical processes and can be spatially separated. In short, resistance arises from the loss of momentum (caused by the hole in this case), while dissipation requires the removal of energy from the electrons. The two involve different physical mechanisms and the elastic or Landauer resistor separates them spatially providing an insightful idealization that makes the physics of resistance much clearer. We argue that a long conductor can be viewed as a series of elastic resistors as sketched in Fig.1.6. and many properties of long conductors, especially at low bias can be understood in simple terms from this viewpoint. Many well-known results like the conductivity and the thermoelectric coefficients for large conductors, that are commonly obtained from the BTE, can be obtained in a more transparent manner by using this viewpoint, as we will show in Part A. We will then use this viewpoint in Part B to look at a variety of quantum transport phenomena like resonant tunneling, conductance quantization, the integer quantum Hall effect and spin precession.

Overview 13

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Fig.1.6. Long resistors can be approximately viewed as a series of elastic resistors, as discussed in Section 3.3. In short, the lesson of nanoelectronics we are trying to convey is the utility of the concept of an elastic resistor with its clean separation of mechanics from thermodynamics. The concept was introduced by Rolf Landauer in 1957 and has been widely used in mesoscopic physics ever since the seminal work in the 1980’s helped establish its relevance to understanding experiments in short conductors. What we hope to convey in these lectures is that the concept of an elastic resistor is not just useful for short conductors but provides a new perspective for long conductors as well, one that makes a wide variety of devices and phenomena transparent and accessible. But why bother with idealizations and approximate physical pictures? Don't we have the BTE and the NEGF equations which provide rigorous frameworks for describing semiclassical and quantum transport respectively? The answer is yes, and all the results we discuss follow directly from the BTE and the NEGF. However, as Feynman (1963) noted in his classic lectures, even when we have an exact mathematical formulation, it is still essential to have an intuitive physical picture:

“.. people .. say .. there is nothing which is not contained in the equations .. if I understand them mathematically inside out, I will understand the physics inside out. Only it doesn’t work that way. .. A physical understanding is a completely unmathematical, imprecise and inexact thing, but absolutely necessary for a physicist.”



Most researchers carry a physical picture in their head and it is usually based on the Drude formula (Eq.(1.5b)). In this book we will show that an alternative picture based on elastic resistors leads to a formula (Eq.(1.5a)) that is more generally valid and argue that it lends more insight into the physics of current flow and the origin of resistance. This picture is widely used in nanoelectronics and mesoscopic physics. We would like to show that its utility goes beyond that: It can also be used to provide an intuitive physical understanding of well-established results for large conductors, results that have been benchmarked against rigorous theories, namely the BTE and the NEGF equations. This viewpoint represents a departure from the established mindset and I hope it will provide a complementary perspective to facilitate the insights needed to take us to the next level of discovery and innovation as we learn to manipulate electrons at a more delicate level involving alternative attributes beyond charge, like spin for example.

Further Reading

In this book we cover many topics using a unique viewpoint that has developed over the years. Some sections of Part A and Part B are based on Datta S. (1995). Electronic Transport in Mesoscopic Systems (Cambridge University Press) Datta S. (2005). Quantum Transport: Atom to Transistor

(Cambridge University Press)

Lessons from Nanoscience: A Lecture Note Series · I: The New Perspective 15 2. Why Electrons Flow...

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